Using the formula Eq. (REF ), we calculate the phonon Hall conductivity
of the kagome lattice systems, the results are shown in Fig. REF . Similar as shown in
Ref. [1]}, we find the nontrivial behavior of the phonon Hall
conductivity as a function of the magnetic field. When \(h\)  is small,
\(\kappa _{xy}\)  is proportional to \(h\) , which is shown in the inset of
Fig.REF ; while the dependence becomes nonlinear when \(h\)  is large.
As \(h\)  is further increased, the magnitude of \(\kappa _{xy}\)  increases before
it reaches a maximum magnitude at certain value of \(h\) .
Then the magnitude of \(\kappa _{xy}\)  decreases and goes to zero at very large \(h\) .
The on-site term \(\tilde{A}^2\)  in the Hamiltonian (REF ) increases
with \(h\)  quadratically so as to blockade the phonon transport, which competes
with the spin-phonon interaction. Because of the coefficient of
\(f(\omega _\sigma )\)  in the summation of the formula Eq. (REF ),
the sign of the Hall conductivity will change with temperatures, which is clearly shown in the inset of Fig.REF . While the phonon hall conductivity at weak magnetic field is always positive for the honeycomb lattice, the sign reverse of the phonon Hall conductivity with temperature for the kagome lattices is novel and interesting, which could be verified by future experimental measurements.
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